Volume 10, Issue 1
Second Order Parameter-uniform Convergence for a Finite Difference Method for a Singularly Perturbed Linear Parabolic System

V. Franklin, M. Paramasivam, J.J.H. Miller & S. Valarmathi


Int. J. Numer. Anal. Mod., 10 (2013), pp. 178-202.

Published online: 2013-10

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  • Abstract

A singularly perturbed linear system of second order partial differential equations of parabolic reaction-diffusion type with given initial and boundary conditions is considered. The diffusion term of each equation is multiplied by a small positive parameter. These singular perturbation parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. Shishkin piecewise-uniform meshes are introduced, which are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that in the maximum norm the numerical approximations obtained with this method are first order convergent in time and essentially second order convergent in the space variable, uniformly with respect to all of the parameters.

  • Keywords

Singular perturbation problems parabolic problems boundary layers uniform convergence finite difference scheme Shishkin mesh

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COPYRIGHT: © Global Science Press

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